Teacher's  Manual 

For 

First  Year  Algebra  Scales 


Hotz 


Published  by 

QTeacfjerg  College,  Columbia  3infter*ftp 

New  York  City 
1922 


Teacher's  Manual 

For 

First  Year  Algebra  Scales 

By 

HENRY  G.  HOTZ,  Ph.D. 

Professor  of  Secondary  Education 
University  of  Arkansas 


Published  by 

Eeadjer*  College,  Columbia  33ntoer*tt|> 

New  York  City 
1922 


Copyright,  1922,  by  Teachers  College,  Columbia  University 


PREFACE 

This  manual  is  compiled  for  the  purpose  of  assisting  teachers 
of  mathematics  in  the  administration  and  practical  use  of  my 
First  Year  Algebra  Scales.  There  is  a  feeling  that  the  original  mono- 
graph, which  appeared  in  the  Teachers  College  Contributions  to 
Education  series,  is  too  technical  and,  consequently,  too  difficult 
for  most  teachers  to  read  intelligently  and  to  determine  from  it 
with  ease  how  to  apply  the  scales  most  profitably.  Suggestions 
concerning  the  purpose  of  these  scales  are  incorporated  in  this 
manual  in  as  simple  and  direct  form  as  possible.  Special  training 
in  statistical  methods  is  not  necessary  for  their  comprehension. 

Besides  the  Tentative  Standards  of  Achievement  proposed  in 
the  original  monograph,  scores  more  recently  obtained  in  various 
cities  and  through  school  surveys  have  been  included.  Suggestions 
on  presentation  and  diagnosis  of  results  have  also  been  added. 
In  order  that  the  progress  of  a  class  may  be  more  scientifically 
determined,  a  revision  of  the  tests,  with  the  exercises  arranged  in 
duplicate  or  alternate  scales  of  equal  difficulty,  will  be  published 
in  the  near  future. 


HENRY  G.  HOTZ 


University  of  Arkansas 
Fayetteville,  Arkansas 


438402 


CONTENTS 

PAGB 

Description  of  the  Scales 7 

Selection  of  Scales  to  be  Used 8 

When  to  Give  the  Tests 9 

Directions  for  Administering  the  Tests 10 

Directions  for  Scoring  the  Papers 12 

Directions  for  Determining  the  Median  or  Class^Score   \    .    .  21 
Standards  of  Achievement 

Tentative  Standard  Scores 26 

Scores  Attained  in  Various  Other  Cities 27 

Graphical    Representation   and   Statistical    Interpretation   of 

Results 34 

Analysis  of  Errors 40 

Value  of  the  Scales 44 

Bibliography 46 


Teacher's  Manual  Fpr 
First  Year  Algebra  Scales 

The  First  Year  Algebra  Scales  were  first  published  in 
Since  then  they  have  been  extensively  used  by  teachers,  school 
administrators,  directors  of  educational  research,  and  in  various 
school  surveys. 

I.    DESCRIPTION  OF  SCALES 

The  scales  consist  of  five  different  sheets  of  algebraic  exercises 
designed  to  measure  the  ability  of  pupils  in  elementary  algebra. 
They  are : 

1.  Addition  and  Subtraction 

2.  Multiplication  and  Division 

3.  Equation  and  Formula 

4.  Graphs 

5.  Problems 

The  first  two  scales,  it  will  be  seen,  are  designed  to  test  the 
achievement  of  students  in  the  fundamental  operations,  involving 
integral,  fractional,  and  radical  expressions;  the  second  two,  to 
test  the  ability  of  students  in  handling  the  instruments  of  quanti- 
tative thinking;  while  the  last  is  composed  of  verbal  problems  of 
the  type  usually  stressed  in  the  first  year  of  algebra. 

The  exercises  in  each  scale  are  arranged  in  order  of  difficulty; 
that  is,  each  scale  begins  with  exercises  so  easy  that  they  can  be 
solved  by  practically  every  member  of  a  class.  Each  succeeding 
exercise,  however,  becomes  increasingly  more  difficult  so  that  the 
last  ones  in  each  scale  can  be  solved  by  only  a  relatively  small 
number  of  students  who  try  them. 

Two  series  of  each  scale  are  offered — Series  A  and  Series  B. 
Series  B  is  the  longer  and  contains  from  eleven  to  twenty-five 
exercises  in  each  scale.  Series  A  is  only  about  one  half  as  long 
and  contains  from  eight  to  twelve  exercises  in  each  scale.  It  covers 
just  as  wide  a  range  of  difficulty  and  has  the  added  advantage  of 
having  the  intervals  between  successive  exercises  and  problems 
approximately  equal:  that  is,  Ex.  3  of  a  given  scale  is  as  much 
more  difficult  than  Ex.  2,  as  Ex.  2  is  more  difficult  than  Ex.  I. 

1Hotz,  Henry  G.:  First  Year  Algebra  Scales.  Teachers  College,  Columbia  Univer- 
sity, Contributions  to  Education,  No.  90. 


8  FIRST  YEAR  ALGEBRA  SCALES 

In  determining  individual  and  class  scores,  the  factor  of  primary 
importance  is  not  so  much  how  many  exercises  an  individual  can 
solve  correctly  in  a  given  time,  but  rather  how  far  along  on  the 
scale  of  exercises,  arranged  in  order  of  increasing  difficulty,  he  can 
perform  satisfactorily.  In  other  words,  the  pupil  is  measured 
almost  entirely  by  the  point  which  he  reaches  on  the  scale.  For 
this  reason  the  tests  may  very  properly  be  technically  called 
"scales,"  and  are  characterized  as  "difficulty  tests"  or  "power  tests" 
by  specialists  in  educational  measurements. 

The  scales  were  derived  from  data  obtained  from  tests  given  to 
over  16,000  high-school  students.  The  schools  which  cooperated 
in  standardizing  the  scales  varied  all  the  way  from  the  small  rural 
high  school  to  the  large  cosmopolitan  high  school.  Classes  were 
tested  in  eighty- four  high  schools  located  in  the  states  of  Massa- 
chusetts, Connecticut,  Rhode  Island,  New  York,  New  Jersey, 
Ohio,  Wisconsin,  Missouri,  Oklahoma,  Colorado,  and  Washington, 
and  the  results  subjected  to  intricate  statistical  treatment.  The 
difficulty  of  each  exercise,  or  its  position  on  the  scale,  was  determined 
by  the  percentage  of  pupils  solving  each  exercise  correctly.2 

2.  SELECTION  OF  SCALES  TO  BE  USED 

Series  A  will  be  found  on  the  whole  to  be  more  satisfactory 
than  Series  B,  especially  where  the  time  available  for  testing 
purposes  is  limited.  This  is  particularly  true  if  the  purpose  of  the 
test  is  primarily  to  determine  degrees  of  attainment.  If,  however,- 
the  purpose  of  the  test  is  mainly  diagnostic,  that  is,  to  discover 
difficulties  which  the  students  are  encountering,  Series  B  should 
be  used.  It  contains  a  richer  variety  of  exercises  and,  consequently, 
a  greater  number  of  type  processes.  This  makes  it  possible  for 
teachers  to  make  a  more  complete  analysis  of  the  mistakes  made 
by  pupils. 

If  only  one  scale  can  be  used,  it  should  be  the  Equation  and 
Formula  Scale,  because  it  is  more  comprehensive  and  so  tests  a 
much  wider  range  of  functions.  At  least  two  scales  should  be 
used,  however,  and  the  scale  which  undoubtedly  comes  second  in 
importance  is  the  Problem  Scale.  If  Series  A  is  used  there  will  be 
ample  time  to  give  both  during  a  single  class  period  of  forty  minutes. 

2  For  a  complete  account  of  the  method  employed  in  locating  each  exercise  on  a  linear 
scale,  consult  Hotz,  Henry  G.:  First  Year  Algebra  Scales. 


WHEN  TO  GIVE  THE  TESTS  9 

Whenever  it  is  possible  to  do  so,  all  five  of  the  tests  of  a  given  series 
should  be  used,  since  the  achievement  on  all  of  the  tests  gives  a 
much  more  reliable  indication  of  a  pupil's  ability  than  the  results 
from  one  or  two  tests  would  give. 

Teachers  have  found  it  most  practicable  to  use  the  tests  in 
rotation  somewhat  as  follows : 

1.  At  the  end  of  three  months 

Addition  and  Subtraction  Scale 
Equation  and  Formula  Scale 

2.  At  the  end  of  six  months 

Multiplication  and  Division  Scale 
Problem  Scale 

3.  At  the  end  of  nine  months 

Equation  and  Formula  Scale  (repeated) 
Graph  Scale 

Whenever  it  is  desired  to  use  the  same  scale  a  second  time,  it 
is  advisable  to  select  it  from  a  different  series.  It  is  feared  that 
unless  at  least  six  months'  time  has  elapsed  since  a  given  test  was 
used  some  of  the  practice  effect  may  survive. 

3.    WHEN  TO  GIVE  THE  TESTS 

The  scales  may  be  used  very  profitably  as  early  as  the  end  of 
the  third  month  of  the  school  year. 

Tentative  Standards  of  Achievement,3  based  upon  the  16,000 
papers  of  the  original  study,  were  compiled  for  three-,  six-,  and 
nine-month  intervals.  It  is,  therefore,  much  more  satisfactory  for 
comparative  purposes  to  submit  these  scales  to  algebra  classes 
immediately  after  they  have  studied  algebra  for  three,  for  six,  or 
for  nine  months.  However,  data  on  the  achievement  at  other 
intervals  are  being  collected  constantly,  much  of  which  is 
included  in  this  pamphlet,4  and  as  time  passes  more  and  more 
information  with  regard  to  the  achievement  that  may  reasonably 
be  expected  at  other  intervals  will  become  available. 

The  scales  are  not  intended  to  be  used  beyond  the  first  year. 
For  this  reason  very  few  results  from  classes  having  had  algebra 
more  than  ten  months  have  been  reported. 

8  See  p.  26. 
4  See  pp.  27-34- 


10  FIRST  YEAR  ALGEBRA  SCALES 

4.    DIRECTIONS  FOR  ADMINISTERING  THE  TESTS 

1 .  Preliminary.    Before  passing  the  papers  see  that  the  desks  are 
cleared  and  pupils  are  provided  with  pencils.  For  the  graph  test  rul- 
ers should  also  be  provided.    Then  make  the  following  statement: 

"I  am  going  to  give  you  a  test  to  see  how  well  you  can 
solve  exercises  in  algebra.  Papers  will  be  passed  to  you 
with  the  printed  side  down.  Please  leave  them  so  until 
I  tell  you  to  turn  them  over." 

2.  Pass  the  papers,  or  have  them  distributed  by  the  pupils  in 
the  front  seats,  with  the  printed  side  down  (Series  B,  first  page 
down)  and  the  top  end  away  from  the  pupil. 

3.  When  all  are  ready  say  to  the  class: 

"Turn  your  papers.  Write  your  name  in  the  first  blank 
space,"  etc.  (The  number  of  blank  spaces  to  be  filled  out 
may  be  determined  by  the  one  giving  the  test.  It  is  not 
necessary  to  have  all  filled  in.  Some  teachers  simply 
have  pupils  write  their  names  in  the  upper  right  hand 
corner  of  the  blank  page  so  as  to  prevent  the  pupils  from 
seeing  any  of  the  exercises  before  all  of  the  directions 
have  been  given.) 

4.  Then  repeat  one  of  the  following  series  of  directions,  depend- 
ing upon  the  tests  to  be  given: 

Addition  and  Subtraction  Scale 

"Attention!  The  exercises  on  these  sheets  are  in  addi- 
tion and  subtraction — collection  of  terms.  Take  the  ex- 
ercises in  the  order  in  which  they  are  given.  Work  as 
many  as  you  can  and  be  sure  you  get  them  right.  Work 
directly  on  these  sheets  and  do  not  ask  anybody  any 
questions.  When  you  have  worked  all  the  exercises  you 
can,  lay  aside  your  pencils  and  remain  quiet  so  as  not 
to  disturb  those  who  are  still  working.  You  will  have 
twenty  minutes  in  which  to  work  (Series  B,  forty  minutes], 
Start." 

Multiplication  and  Division  Scale 

"Attention!  The  exercises  on  these  sheets  are  in  multi- 
plication and  division.  Take  the  exercises  in  the  order  in 
which  they  are  given.  Work  as  many  as  you  can  and  be 


DIRECTIONS  FOR  ADMINISTERING  THE  TESTS  1 1 

sure  you  get  them  right.  All  answers  must  be  reduced  to 
their  simplest  forms.  Work  directly  on  these  sheets  and 
do  not  ask  anybody  any  questions.  When  you  have  worked 
all  the  exercises  you  can,  lay  aside  your  pencils  and  re- 
main quiet  so  as  not  to  disturb  those  who  are  still  work- 
ing. You  will  have  twenty  minutes  in  which  to  work 
(Series  B,  forty  minutes).  Start." 

Eqiiation  and  Formula  Scale 

"Attention!  On  these  sheets  you  are  given  a  number 
of  equations  and  formulae  to  solve.  Take  the  exercises 
in  the  order  in  which  they  are  given.  Solve  as  many  as 
you  can  and  be  sure  you  get  them  right.  Work  directly 
on  these  sheets  and  do  not  ask  anybody  any  questions. 
When  you  have  worked  all  the  exercises  you  can,  lay  aside 
your  pencils  and  remain  quiet  so  as  to  not  disturb  those 
who  are  still  working.  You  will  have  twenty  minutes  in 
which  to  work  (Series  B,  forty  minutes).  Start." 

Graph  Scale 

"Attention!  On  these  sheets  you  are  given  a  number 
of  graphs.  Read  each  question  carefully  and  then  do 
as  you  are  told  to  do.  Take  the  exercises  in  the  order  in 
which  they  are  given.  Answer  as  many  questions  as  you 
can  and  be  sure  you  get  them  right.  Work  directly  on 
these  sheets  and  do  not  ask  anybody  any  questions. 
When  you  have  answered  all  the  questions  you  can,  lay 
aside  your  pencils  and  remain  quiet  so  as  not  to  disturb 
those  who  are  still  working.  You  will  have  twenty-five 
minutes  in  which  to  work.  Start." 

Problem  Scale 

"Attention!  On  these  sheets  you  are  given  a  number 
of  questions  to  answer.  Take  the  exercises  in  the  order 
in  which  they  are  given.  Answer  as  many  questions  as 
you  can  and  be  sure  you  get  them  right.  In  all  the  prob- 
lems which  call  for  the  equation,  for  example  No.  4, 
simply  state  the  equation  which  will  solve  the  problem. 
Take  for  example,  this  problem:  A  coat  and  hat  cost 
$30.  The  coat  cost  5  times  as  much  as  the  hat.  Find 
the  cost  of  each.  The  equation  would  be  x  +  $x  —  $30. 


12  FIRST  YEAR  ALGEBRA  SCALES 

(Write  the  equation  on  the  board.)  Work  directly  on 
these  sheets  and  do  not  ask  anybody  any  questions. 
When  you  have  answered  all  the  questions  you  can, 
lay  aside  your  pencils  and  remain  quiet  so  as  not  to  dis- 
turb those  who  are  still  working.  You  will  have  twenty-five 
minutes  in  which  to  work  (Series  B,  forty  minutes).  Start." 

5.  When  the  time  is  up,  say  "Stop"  and  collect  the  papers. 
Most  of  the  pupils  will  have  finished  before  this  time.  Those  who 
have  not,  in  all  probability  have  done  all  they  can. 

A  warning,  stating  the  amount  of  time  left,  should  be  given 
three  minutes  before  time  is  called  for  the  tests  of  Series  A  and  five 
minutes  in  advance  for  those  of  Series  B. 

With  many  classes  which  have  had  less  than  nine  months  of 
algebra,  and  especially  those  which  have  had  only  three  months,  it 
is  perfectly  safe  to  call  time  before  the  full  time  allowed  for  that 
particular  test  has  elapsed. 

Students  may  be  provided  with  scratch  paper  for  their  own  use. 
It  has  been  found  to  be  most  satisfactory  to  pass  quietly  down  the 
aisles  a  few  minutes  after  the  test  is  started  and  give  each  pupil 
a  sheet  of  scratch  paper.  Pupils  will  find  it  more  convenient,  how- 
ever, to  work  directly  on  the  question  sheets.  For  all  but  the 
problem  test  it  is  desirable  to  have  as  much  of  the  work  as  possible 
on  these  sheets. 

5.    DIRECTIONS  FOR  SCORING  THE  PAPERS 

In  scoring  the  papers,  answers  are  to  be  marked  either  right  or 
wrong.  All  answers  which  may  be  accepted  as  correct  are  given 
on  pages  13  to  19  of  this  manual.  A  few  incorrect  answers  are 
also  listed,  to  indicate  more  definitely  the  types  of  answers 
which  must  not  be  accepted. 

No  credit  is  given  for  answers  that  are  partially  right.  This 
procedure,  though  somewhat  arbitrary,  greatly  simplifies  the  task 
of  scoring  the  papers.  It  saves  time  and  offers  less  chance  for 
variation  in  scoring  the  results.  This  last  factor,  uniformity  in 
scoring,  is  an  absolute  essential  in  order  that  valid  comparisons 
between  different  school  systems  may  be  made. 

After  each  paper  is  scored,  it  has  been  found  most  convenient 
to  record  the  total  number  of  exercises  solved  correctly  in  the 
upper  right-hand  corner  of  the  test  sheet. 


DIRECTIONS  FOR  SCORING  THE  PAPERS 
ANSWER  KEY  TO  ADDITION  AND  SUBTRACTION  SCALE 


Problem 
No. 

Answers 

Problem 

No. 

Answers 

I 

9? 

2rj            2^2 

2 
? 

5* 
156 

r2^2.(,_2)(,+2) 

4 

9-6*    9       6* 

5 

6x  +2 

4     '  4        4 

6 

7 

1 

5-3<* 

8 

5* 

20 

(a  +  i)  (a*-a+i)' 

9 

10 

5<  +  6 

40-13* 

ii 

a2  -66  +4 

21 

40-13* 

12 

4*-  1 
3c   6c 

*3-i9*+30 
I                     i 

15 

8'  16 

6 

a  —  2* 

22 
23 
24 

5iVs;  n°t  5  Vs  +  s  Vs 

6g-7.  7^6a 
a2  —  4  '  4  —  a2 

14  FIRST  YEAR  ALGEBRA  SCALES 

ANSWER  KEY  TO  MULTIPLICATION  AND  DIVISION  SCALE 


Problem 
No. 

Answers 

Problem 
No. 

Answers 

I 

2iy 

14 

4*-  8;  40  -2) 

2 

3" 

•r  ^ 

&               6 

3 

8a262 

J5 

a(m  —  n)  '  am—  an 

4 

30 

16 

si*y 

5 

6m 

c2     <f2 

6 

-2b 

17 

/"     1     /-7  .  *i  /~\4" 

6  n^  a-  ,  IIUL            j 
c  —  a 

4 

7 

-I2*2/ 

18 

I2x2;  I2^2 

c 

6a5 

O 

T  *"V 

I   2a  —  i 

9 

2m  —  $n 

19 

2'        2 

10 

2X* 

-2O 

^>  —  2            p  —  2 

5 

y(P~9Y  3pr-27 

ii 

ioa3  +  33a2-52a  +  9 

/^  T 

*/2  —  -?  V  _1_  Q     -v-^  —  'IX  -\-  Q 
•^         o*^     I     V  .  *         O**'     1     V 

12 

w2—  10 

21 

3  \^     3/           3*^     9 

22 

48 

13 

_4. 

3a 

23 

3 

DIRECTIONS  FOR  SCORING  THE  PAPERS 
ANSWER  KEY  TO  EQUATION  AND  FORMULA  SCALE 


Problem 
No. 

Answers 

Problem 
No. 

Answers 

I 
2 

2 

3 

15 

I 
3 

3 

2 

16 

154 

4 

5 

7 

2 

17 

El 
R 

6 

2;  not  —2=  —2 

18 

~;not^; 

7 

9 

i 
nor  —  x=- 

2 

8 

2 

19 

10,  —  5  (both  roots) 

9 

4 

20 

-5,  -i*;  not  -*  =  5 

10 

15 

16 

21 

,-,,,-4-          . 

ii 

12 

40 

2  .30 

47'7 

22 
23 

«;•? 

i 

13 

15 

24 

is.  rr.i 

\j  Vir  r^ 

14 

m=2,n  =  4* 

25 

1 

*  Where  two  results  are  ordinarily  required  in  an  answer,  the  exercise  is  marked 
correct  if  the  work  is  done  correctly  up  to  the  point  where  only  one  value  is  obtained 
and  stopped  at  that.  If,  however,  the  student  makes  an  error  in  solving  for  the  second 
value,  the  problem  is  scored  incorrect. 


16 


FIRST  YEAR  ALGEBRA  SCALES 

ANSWER  KEY  TO  PROBLEM  SCALE 


Problem 
No. 

Answers 

I 

3* 

2 

m  —  r 

3 

a  +  b 

4 

X  +  IOX 

=  132.  ($12  and  $120) 

5 

980 

V 

6 

4*  +  40 

=  240;  2x  -|-  2(x  —  20)  =  240;  4^  -  40  =  240. 

(50  ft. 

x  70  ft.) 

7 

3*  -  136  =  836;  2x  —  136  =  836  -  #;  a;  +  y  =  836 

and#  = 

=  2y  —  136.  (324  children  and  512  adults) 

8 

-T-;  not 
dw 

dwx  =  r 

9 

(X  +  12) 

(x  -  4)  =  x2.  (6  ft.  X  6  ft.  and  2  ft.  X  18  ft.) 

10 

2i:5f  = 

20  :  x  ;  x  :  20  —  5  ft.  9  in.  :  2  ft.  6  in.  ;  x  :  20  = 

69  :30 

;  eight  times  as  high  as  the  man;  not  2.6  :  5.9  = 

20  :  x. 

(46  ft.) 

ii 

x  +  y  = 

5000  and  —  +  -42  =  ^2; 

100         100 

3*   + 

4.(=;ooo  —  x} 

—   T*?7?*    ri'iv  —1-  onn  ••-•    r»  1  v  —  fTy 

100 

IOO              :72>  -OS*   I  20(        -°4^  -    J72- 

($2800 

and  $2200) 

12 

40*  =  5^ 

5(*  -  2);  55*  =  40(*  +  2);  -«  -  2  =  — 

(2931A 

miles)                                       40                55 

13 

x  +  y  = 

20  and  50^  +  65^  =  1200;  50^  +  65(20  —  x)  = 

1200;  not  SQX  +  65(20  —  x)  =  12.   (6%  Ibs.  and  13^3 

Ibs.) 

14 

5*2  =  i* 

D;5(#  —  io)2  =  180.  (i6in.  X  i6in.) 

The  equations  given  above  are  those  which  are  usually  found.    Modific?tions, 
which  in  the  end  equal  the  same,  may  be  accepted.  For  example,  4*  =  240  +  40  is  the 

same  as  4*  —  40  =  240,  and  —  =  —  is  the  same  as  69  : 30  =  x  :  20.    Where  the  prob- 

30     20 

lems  have  been  worked  out  and  the  correct  answers  are  given,  they  are  to  be  scored 
as  correct,  though  such  a  procedure  on  the   part  of  students  is  to  be  discouraged. 


DIRECTIONS  FOR  SCORING  THE  PAPERS 
ANSWER  KEY  TO  GRAPH  SCALE 


Problem 
No. 


1 .  2000 

2.  24 

3-          3330  to  3350 
(inclusive) 


5- 

6. 

and 
9- 


-    «,. 


11 


-3  to  -9 
(inclusive) 


TEARS 


•      »       *       6       a      10     la     14      ia      ia     ao 


18 


FIRST  YEAR  ALGEBRA  SCALES 


7- 


\ 


1  Spaos  •  l-  unit 


8. 


72. 


Sb 


'to 


32 


3 
FEET 


IO. 


DIRECTIONS  FOR  SCORING  THE  PAPERS 

x  =  3 
y  =  2. 


/  r 


1  Space  •  1  Unit 


ii.         In  twelve  weeks;  the  thirteenth  week. 


3  s 


f       fa      8      10     12.     m     it 

WEEKS 


20 


FIRST  YEAR  ALGEBRA  SCALES 


The  score  assigned  to  each  pupil  for  each  test  is  the  total  number 
of  "rights,"  that  is,  the  total  number  of  exercises  solved  correctly 
in  the  test.  Individual  scores  made  on  the  various  tests  by  the 
different  pupils  of  a  class  may  be  recorded  on  a  sheet  similar  to 
Fig.  I. 


City 

School..  Teacher. 


Date 


Grade. 


Score 

s  on  Each 

Test 

I 

II 

III 

IV 

V 

Median  Score 

Standard  Score 

FIG.  i 

Individual  Class  Record  Sheet  Used  in  Recording  the  Results  for  Each  Class 


DETERMINING  THE  MEDIAN  OR  CLASS  SCORE  21 

6.     DIRECTIONS    FOR    DETERMINING    THE    MEDIAN    OR    CLASS    SCORE 

The  median  number  of  exercises  correctly  solved  is  used  in 
connection  with  these  scales  as  the  class  score.  Though  not  entirely 
accurate  scientifically,  it  is  the  most  readily  computed  and  is, 
therefore,  for  all  practical  purposes  the  most  satisfactory  measure 
of  the  achievement  of  a  class.  The  median  score  represents  the 
number  of  exercises  solved  correctly  by  just  fifty  per  cent  of  a 
class.  That  is,  there  are  just  as  many  students  in  a  class  who  solve 
a  larger  number  as  there  are  students  who  solve  a  smaller  number 
of  exercises. 

In  order  to  determine  the  median  point  of  the  achievement  of  a 
class,  it  is  necessary  to  make  a  distribution  table  of  the  results 
of  a  test.  Such  a  table  shows  the  number  of  pupils  who  were  unable 
to  solve  a  single  exercise  correctly,  the  number  who  solved  one 
exercise  correctly,  two  exercises,  three  exercises,  etc.  Sample  dis- 
tributions for  four  of  the  tests,  Addition  and  Subtraction,  Multi- 
plication and  Division,  Equation  and  Formula,  and  Problems,  are 
shown  in  Table  I,  (page  22).  Another  distribution  is  given  in 
Table  II,  (page  23). 


22 


FIRST  YEAR  ALGEBRA  SCALES 


TABLE  I 

DISTRIBUTION  TABLE  SHOWING  SCORES  ATTAINED  FOR  FOUR  OF  THE 
TESTS,  OKMULGEE,  OKLAHOMA  6 


City     Okmulgee 

School     High 

Remarks    9-month  group 


State     Oklahoma 
Teacher 


Date     June  1921 
.  Grade     pth 


Number  of  Pupils  Making  Score  Indicated 

No   of 

IMvy.    Ul 

Examples 

Addition 

Multiplica- 

Equation 

Correct 

and 

tion  and 

and 

Problems 

Graphs 

Subtraction 

Division 

Formula 

25 

24 

23 

22 

21 

2O 

19 

18 

17 

16 

15 

14 

13 

12 

6 

6 

12 

. 

II 

5      ' 

8 

12 

; 

10 

17 

9 

12 

2 

9 

7<5 

9 

12 

6 

^  3 

8 

-r/7 

21 

19 

ii 

41 

7 

ii 

14 

9 

10 

•44 

6 

7 

12 

9 

16 

U^ 

5 

7 

9 

7 

19 

4* 

4 

5 

6 

2 

18 

3  i 

3 

4 

4 

2 

12 

•;  '*• 

2 

2 

4 

t, 

I 

O 

Total 

97 

98 

96 

98 

Median 

8.74 

8.19 

9.0 

5-79 

Standard 

7-9 

7-9 

7.8 

5-6 

Form  used  by  Bureau  of  Educational  Research,  University  of  Illinois,  Urbana,  111. 


DETERMINING  THE  MEDIAN  OR  CLASS  SCORE  23 

Table  II  also  indicates  in  a  clear  and  concise  way  the  method 
of  computing  the  median  class  score.6  Since  a  thorough  knowledge 
of  the  method  of  calculating  the  median  is  essential  to  the  proper 
use  of  these  scales,  it  is  urged  that  teachers  who  are  not  familiar 
with  educational  statistics  make  a  careful  study  of  this  table. 


TABLE  II 

SAMPLE  DISTRIBUTION  OF  SCORES  MADE  ON  EQUATION  AND  FORMULA 
TEST,  SERIES  B 


Score 

Number  of  Pupils 
Making  Each  Score 

Computation  of  Median 

20 

i.  After  checking  the  problems  correctly 

19 

solved,  count  the  check  marks  in  each 

18 

paper,  and  indicate  the  number  in  the 

17 

II 

2 

proper  place  in  column  2. 

16 

1111 

4 

2.  Find  the  total  number  of  scores  (N). 

15 

1  1  i 

3 

. 

14 

Mill 

5 

3.  Median    equals    middle    score  =  — 

13 

Illll  II 

7 

2 

12 

III 

3 

II 
IO 

Illl 
Ill 

4 
3 

N       is 
Thus,  —  =  ^  =  i7# 

9 

II 

2 

'    2            2 

.     8 

Beginning  at  i  in  the  third  column 

6 

5 

1 
1 

I 
I 

and  counting  up,  it  is  necessary  to 
count  3>^  of  the  7  to  make  17^,  thus: 

4 

* 

171A  =  1  +  1+2  +  3+4+3  and 

3 

2 

3^  of  the  7. 

I 

Put  3^  as  a  numerator  over  7  and 

O 

add  to  13,  the  step  on  which  the  7 

occurs,  thus: 

Total  Scores  (N) 

35 

Median  =  13  +  ^  =  13.5 

Median  Score 

13-5 

Errors  in  computing  the  median  or  class  score  are  very  common ; 
and,  when  comparison  is  to  be  made  with  standard  scores,  an  error 
of  one-half  point  in  its  computation  may  do  serious  injustice  to 
a  class.  As  a  further  safeguard  against  error  in  this  important  de- 
tail, four  additional  class  distributions  are  submitted  in  Table  III, 
and  a  complete  discussion  of  the  method  of  determining  the  median 
or  class  score  of  these  distributions  follows. 

6  The  method  suggested  here  is  an  adaptation  of  the  plan  used  by  Clifford  Woody  in 
his  The  Woody  Arithmetic  Scales  and  How  To  Use  Them,  p.  19. 


24  FIRST  YEAR  ALGEBRA  SCALES 

TABLE  III 

SAMPLE   DISTRIBUTION  OF  SCORES  MADE  BY  FOUR   DIFFERENT  CLASSES  ON 
EQUATION  AND  FORMULA  TEST 


Number  of 
Problems 
Solved 

Class 
I 

Class 
II 

Class 
III 

Class 
IV 

o 

i 

I 

2 

2 

2 

i 

I 

i 

3 

2 

2 

3 

4 

I 

I 

2 

5 

4 

I 

2 

3 

6 

6 

5 

3 

2 

7 

3 

4 

I 

8 

3 

3 

I 

3 

9 

2 

2 

2 

4 

10 

I 

I 

I 

2 

ii 

I 

I 

I 

12 

13 

I 

14 

I 

15 

Total 

26 

19 

16 

22 

Median  Score 

6.5 

7.6 

6.0 

7-5 

According  to  this  table,  there  were  twenty-six  students  in  Class  I, 
nineteen  students  in  Class  II,  sixteen  students  in  Class  III,  and 
twenty-two  students  in  Class  IV.  In  Class  I  one  student  solved 
one  problem  correctly,  two  solved  two  problems  correctly,  two 
solved  three  correctly,  etc.  To  find  the  median  score  of  this  class, 
it  is  necessary  to  find  the  point  in  the  distribution  of  the  class  where 
there  are  just  as  many  students  who  solved  a  greater  number  of 
problems  as  there  are  students  who  solved  a  smaller  number. 
Since  there  are  twenty-six  students  in  the  class,  this  point  is  obvi- 
ously midway  between  the  scores  made  by  the  thirteenth  and 
fourteenth  students,  counting  down  in  the  distribution.  That  is, 
to  include  the  thirteenth  individual  with  the  poorer  group,  it  is 
necessary  to  count  three  of  the  six  students  who  solved  six  prob- 
lems; and,  since  it  is  assumed  that  the  individuals  are  distributed 


STANDARDS  OF  ACHIEVEMENT  25 

evenly  through  a  step  at  equal  distances  from  one  another,  the 
median  point  is  just  one  half  of  the  distance  through  this  step,  from 
six  to  seven.  Therefore,  the  median  score  of  this  class  is  6.5  problems 
solved  correctly. 

In  Class  II  there  are  nineteen  students.  Thus,  there  are  9.5 
individuals  both  above  and  below  the  exact  median  point  in  the 
distribution  of  this  class.  To  include  9.5  individuals  with  the 
poorer  group,  it  is  necessary  to  count  2.5  of  the  4  students  who 

solved  7  problems.  Hence,  the  median  point  is  just  -     -   of   the 

4 

distance  through  the  7th  step,  which  makes  the  median  score  of 
this  class  7.6  problems  solved  correctly. 

Class  III  illustrates  another  situation  still.  There  are  16  students 
in  the  class,  and  in  counting  out  the  8  individuals  for  the  poorer 
group,  we  exactly  take  up  all  the  cases  in  the  5th  step.  The  fact 
to  be  noticed  here  is  that  the  median  point  is  raised  clear  through 
the  5th  step.  The  median  score  for  this  class,  therefore,  is  6.0 
problems  correctly  solved. 

Class  IV  is  here  included  to  assist  in  the  solution  of  another  dif- 
ficulty which  is  often  encountered.  There  are  22  individuals  in  the 
class.  Counting  from  the  top  of  the  distribution,  the  1 1  cases  for 
the  poorer  group  take  us,  as  seen  above,  entirely  through  step  6. 
Likewise,  counting  from  below,  to  include  n  cases,  we  have  to  go 
clear  through  step  8.  From  this  it  appears  that  the  median  point 
could  be  located  all  the  way  from  7  to  8  in  the  class  distribution. 
Since,  however,  any  given  distance  on  a  scale  is  best  represented 
by  its  middle  point,  the  median  score  of  this  class  should  be  7.5 
problems  solved  correctly. 

7.    STANDARDS  OF  ACHIEVEMENT 

Tentative  standards  of  achievement  were  proposed  in  the  orig- 
inal monograph.7  When  these  were  published,  the  scales  had  not 
been  used  very  extensively,  and  some  doubt  was  expressed  with 
regard  to  the  reliability  of  the  tentative  standards.  There  is  as 
yet,  however,  no  conclusive  evidence  that  any  of  these  standards 
should  be  materially  revised. 

There  is  some  evidence  that  the  tentative  standards  are  on  the 
whole  a  little  too  high.  On  the  other  hand,  whenever  the  tests 

7  Hotz,  Henry  G:   First  Year  Algebra  Scales,  p.  41. 


26 


FIRST  YEAR  ALGEBRA  SCALES 


have  been  submitted  to  classes  in  large  high  schools,  where  the 
teaching  is  generally  more  efficient  and  where  there  is  more  careful 
selection  of  the  subject  matter  taught  in  elementary  algebra,  the 
results  invariably  surpass  these  standards. 


TABLE  IV 
TENTATIVE  MEDIAN  STANDARDS  OF  ACHIEVEMENT,  SERIES  A 


Three-Month 
Group 

Six-  Month 
Group 

Nine-Month 
Group 

Addition  and  Subtraction 

5-o 

6.8 

7-9 

Multiplication  and  Division 

5-3 

6-3 

7-9 

Equation  and  Formula 

4-9 

7-i 

7-8 

Problem  Test 

4-3 

4.9 

5-6 

Graph  Test 

2.8  (four  and  one-half 

5-6 

months) 

TABLE  V 
TENTATIVE  MEDIAN  STANDARDS  OF  ACHIEVEMENT,  SERIES  B 


Three-Month 
Group 

Six-Month 
Group 

Nine-Month 
Group 

Addition  and  Subtraction 

9-7 

12.9 

14.4 

Multiplication  and  Division 

9.6 

14.0 

16.3 

Equation  and  Formula 

7.8 

14-3 

16.0 

Problem  Test 

54 

6-5 

7-5 

Graph  Test 

3.7  (four  and  one-half 

7.2 

months) 

STANDARDS  OF  ACHIEVEMENT 


TABLE  VI 
SUMMARY  OF  MEDIAN  SCORES  ATTAINED  IN  VARIOUS  CITIES  WITH  FIRST  YEAR 

ALGEBRA  SCALES,  SERIES  A 
ADDITION  AND  SUBTRACTION  SCALE 


City 

No.  of  Months  of  Algebra  Studied 

3 

6 

8 

9 

16 

14 

Illinois  Cities 

6-9 

7-3 

Rural  Schools 

4-3 

4-9 

Virginia     Small  City  Schools 

5-2 

5-9 

Large  City  Schools 

3-6 

5-3 

5-o 

North  1  Rural  Sch°°ls 

3-3 

3-7 

4-5 

4.9 

'     ,        \  Small  City  Schools 
Carolina     T          ~.     c  ,      , 
J  Large  City  Schools 

2.9 
3-6 

5-4 
3-9 

3.9 

4-3 

54 

Atlantic  City,  N.  J. 

7-4 

Reading,  Pa. 

3-9 

Providence,  R.  I. 

(Moses  Brown  School) 

6-7 

10.5 

Fayetteville,  Ark. 

(University  H.  S.) 

5-i 

5-3 

7-5 

Fayetteville,  (City  H.  S.) 

5-3 

Little  Rock,  Ark. 

4-9 

6.0 

Potsdam,  N.  Y. 

6-3 

Saratoga,  N.  Y. 

54 

Elmira,  N.  Y. 

6.1 

Whitehall,  N.  Y. 

6.9 

Cities  of  Original  Study 

5-0 

6.8 

7-9 

MULTIPLICATION  AND  DIVISION  SCALE 

No.  of  Months  of  Algebra  Studied 

City 

3 

6 

8 

9 

IO 

14 

Illinois  Cities 

7.2 

74 

Athens,  Ohio 

3-8 

4-8 

5-2 

Reading,  Pa. 

3-9 

Providence,  R.  I. 

(Moses  Brown  School) 

7-7 

10.4 

Fayetteville,  Ark. 

(University  H.  S.) 

5-6 

5-9 

6.8 

Lockport,  N.  Y. 

6.7 

Potsdam,  N.  Y. 

5.8 

Saratoga,  N.  Y. 

6.2 

Elmira,  N.  Y. 

5.9 

Amsterdam,  N.  Y. 

M. 

Cities  of  Original  Study 

5-3 

6-3 

7-9 

28 


FTRST  YEAR  ALGEBRA  SCALES 


TABLE  VI 
SUMMARY  OF  M  EDIAN  SCORES  ATTAINED  IN  VARIOUS  CITIES  WITH  FIRST  YEAR 

ALGEBRA  SCALES,  SERIES  A 
EQUATION  AND  FORMULA  SCALE 


No.  of  Months  of  Algebra  Studied 

ity 

3 

6 

8 

9 

10 

14 

Illinois  Cities 

77 

7-9 

1  Rural  Schools 

1.6 

3-9 

Virginia  \  Small  City  Schools 

4.6 

5-8 

J  Large  City  Schools 

4.0 

54 

5-1 

)  Rural  Schools 

3-2 

4.2 

47 

4.9 

North  Carolina  >  Small  City  Schools 

3.1 

2.8 

1.5 

5.1 

J  Large  City  Schools 

3-3 

4-5 

5-5 

Reading,  Pa. 

1.8 

Atlantic  City,  N.  J. 

6.2 

Chicago,  111. 

(Research  Study  by  Eleanora  Harris) 

6-3 

6-9 

Providence,  R.  I. 

(Moses  Brown  School) 

9.0 

9.8 

Fayetteville,  Ark. 

(University  H.  S.) 

5-9 

6-3 

8.6 

Philadelphia,  Pa. 

6.8 

Little  Rock,  Ark. 

6.0 

6.8 

Lockport,  N.  Y. 

7-3 

Potsdam,  N.  Y. 

6.4 

Saratoga,  N.  Y. 

54 

Whitehall,  N.  Y. 

7.8 

Amsterdam,  N.  Y. 

8.1 

Cities  of  Original  Study 

4.9 

7-1 

7-8 

STANDARDS  OF  ACHIEVEMENT 


29 


TABLE  VI 

SUMMARY  OF  MEDIAN  SCORES  ATTAINED  IN  VARIOUS  CITIES  WITH  FIRST  YEAR 

ALGEBRA  SCALES,  SERIES  A 

GRAPH  SCALE 


No.  of  Months  of  Algebra  Studied 

City 

4X 

6 

8 

9 

10 

12 

14 

Chicago,    111. 

(Research    Study    by 

Eleanora  Harris) 

4-7 

Fayetteville,  Ark. 

(University  H.  S.) 

4-5 

6.0 

Providence,  R.  I. 

(Moses  Brown  School) 

6.5 

6.8 

Reading,  Pa. 

1.8 

Illinois  Cities 

6.2 

5-0 

Wellington,  Kans.  (1919) 

6.6 

Wellington,  Kans.  (1920) 

6.4 

Wellington,  Kans.  (1921) 

Rapid  Group 

7.8 

Average  Group 

4.8 

Slow  Group 

4.6 

Mississippi 

(Two  Schools) 

4.2 

South  West  City,  Mo. 

Hackensack,  N.  J. 

2-3 

5-6 

5-2 

7-3 

74 

Amsterdam,  N.  Y. 

2-5 

Cities  of  Original  Study 

2.8 

5-6 

FIRST  YEAR  ALGEBRA  SCALES 


TABLE  VI 

SUMMARY  OF  MEDIAN  SCORES  ATTAINED  IN  VARIOUS  CITIES  WITH  FIRST  YEAR 

ALGEBRA  SCALES,  SERIES  A 

PROBLEM  SCALE 


No.  of  Months  of  Algebra  Studied 

City 

3 

6 

8 

9 

10 

H 

Illinois  Cities 

6.4 

5-0 

Athens,  Ohio 

1.9 

2-5 

3-8 

Reading,  Pa. 

2.O 

Chicago,  111. 

(Research  Study  by 

Eleanora  Harris) 

4-3 

Providence,  R.  I. 

(Moses  Brown  School) 

5-8 

7.2 

Mount  Holly,  N.  J. 

3-6 

Fayetteville,  Ark. 

(University  H.  S.) 

4-3 

4-3 

6.2 

Saratoga,  N.  Y. 

4.2 

Elmira,  N.  Y. 

4-9 

Amsterdam,  N.  Y. 

5.5 

Cities  of  Original  Study 

4-3 

4-9 

5-6 

STANDARDS  OF  ACHIEVEMENT 


TABLE  VII 
SUMMARY  OF  MEDIAN  SCORES  ATTAINED  IN  VARIOUS  CITIES  WITH  FIRST  YEAR 

ALGEBRA  SCALES,  SERIES  B 
ADDITION  AND  SUBTRACTION  SCALE 


No.  of  Months  of  Algebra  Studied 

City 

3 

6 

8 

9 

10 

12 

14 

Cleveland,  Ohio 

14.7 

Fayetteville,  Ark. 

• 

(University  H.  S.) 

94 

New  Jersey  Cities 

I2.O 

Racine,  Wis.  (1919) 

13-0 

17-3 

Racine,  Wis.  (1920) 

13-6 

18.8 

Wisconsin  Cities  (1918) 

II.  2 

14.4 

Wisconsin,  22  Cities  (1921) 

14.8 

Wellington,  Kans.  (1919) 

18.1 

Wellington,  Kans.  (1920) 

15-9 

Wellington,  Kans.  (1921) 

Rapid  Group 

19-5 

Average  Group 

14.1 

Slow  Group 

10.8 

Elizabeth  City,  N.  J. 

12.8 

Andover,  Mass. 

10.0 

South  West  City,  Mo. 

17.0 

Hackensack,  N.  J. 

12.0 

14-5 

18.0 

2O.O 

Paragould,  Ark. 

10.8 

12.6 

12.  1 

Cities  of  Original  Study 

9-7 

12.9 

14.4 

FIRST  YEAR  ALGEBRA  SCALES 


TABLE  VII 
SUMMARY  OF  MEDIAN  SCORES  ATTAINED  IN  VARIOUS  CITIES  WITH  FIRST  YEAR 

ALGEBRA  SCALES,  SERIES  B 
MULTIPLICATION  AND  DIVISION  SCALE 


No.  of  Months  of  Algebra  Studied 

City 

3 

6 

8 

9 

10 

12 

H 

Cleveland,  Ohio 

14.8 

New  Jersey  Cities 

154 

Racine,  Wis.  (1919) 

13.6 

18.1 

Racine,  Wis.  (1920) 

14.7 

18.6 

Waukesha,  Wis. 

14.0 

Wisconsin  Cities  (1918) 

11.4 

17.4 

Wisconsin,  23  Cities  (1921) 

16.2 

Wellington,  Kans.  (1919) 

18.3 

Wellington,  Kans.  (1920) 

17.4 

Wellington,  Kans.  (1921) 

Rapid  Group 

17.0 

Average  Group 

14.0 

Slow  Group 

I3-I 

Andover,  Mass. 

14.2 

Ironwood,  Mich. 

II.  0 

Hackensack,  N.  J. 

10.4 

15-9 

16-5 

19-5 

Paragould,  Ark. 

H-3 

I3-I 

14.9 

Cities  of  Original  Study 

9.6 

14.0 

16-3 

STANDARDS  OF  ACHIEVEMENT 


33 


TABLE  VII 
SUMMARY  OF  MEDIAN  SCORES  ATTAINED  IN  VARIOUS  CITIES  WITH  FIRST  YEAR 

ALGEBRA  SCALES,  SERIES  B 
EQUATION  AND  FORMULA  SCALE 


No.  of  Months  of  Algebra  Studied 

City 

3 

.  6 

8 

9 

10 

12 

14 

Cleveland,  Ohio 

14.9 

Fort  Smith,  Ark. 

7.2 

9-9 

New  Jersey  Cities 

16.3 

Racine,  Wis.  (1919) 

17-5 

Racine,  Wis.  (1920) 

15-9 

Wisconsin  Cities  (1918) 

17.2 

Wisconsin,  28  Cities  (1921) 

14.2 

Wellington,  Kans.  (1919) 

18.1 

Wellington,  Kans.  (1920) 

16.7 

Wellington,  Kans.  (1921) 

Rapid  Group 

20.  6 

Average  Group 

14.4 

Slow  Group 

II.2 

Elizabeth  City,  N.  C. 

12.8 

Andover,  Mass. 

12.8 

Ironwood,  Mich. 

6-4 

Cold  Springs,  N.  Y. 

(Holdane  School) 

13-5 

Hackensack,  N.  J. 

9.0 

14.5 

2O.O 

20.3 

Paragould,  Ark, 

II.O 

12.5 

15-8 

Cities  of  Original  Study 

7.8 

14-3 

16.0 

34 


FIRST  YEAR  ALGEBRA  SCALES 


TABLE  VII 

SUMMARY  OF  MEDIAN  SCORES  ATTAINED  IN  VARIOUS  CITIES  WITH  FIRST  YEAR 

ALGEBRA  SCALES,  SERIES  B 

PROBLEM  SCALE 


No.  of  Months  of  Algebra  Studied 

City 

3 

6 

8 

9 

10 

12 

14 

Cleveland,  Ohio 

6-3 

Fort  Smith,  Ark. 

4-7 

5-0 

New  Jersey  Cities 

7.0 

Waukesha,  Wis. 

7-9 

Wisconsin  Cities  (1918) 

8.2 

Wisconsin,  31  Cities  (1921) 

7-i 

Wellington,  Kans.  (1919) 

9.8 

Wellington,  Kans.  (1920) 

94 

Wellington,  Kans.  (1921) 

Rapid  Group 

11.4 

Average  Group 

7.8 

Slow  Group 

5-8 

Elizabeth  City,  N.  C. 

8.1 

South  West  City,  Mo. 

8-3 

Mississippi  (Two  Schools) 

4-7 

6-3 

Cold  Springs,  N.  Y. 

(Holdane  School) 

5-8 

Hackensack,  N.  J. 

5-5 

5-3 

5-9 

54 

Paragould,  Ark. 

7.0 

6.1 

74 

Cities  of  Original  Study 

54 

6-5 

7-5 

8.    GRAPHICAL  REPRESENTATION  AND   STATISTICAL  INTERPRETATION 

OF    RESULTS 

After  the  papers  for  one  or  more  of  the  tests  of  a  class  have 
been  scored,  and  the  median  has  been  computed,  the  results  should 
be  entered  in  tabular  form  on  an  Individual  Class  Record  Sheet 
similar  to  the  one  shown  in  Fig.  I  (p.  20). 

In  large  school  systems  it  is  usually  necessary  to  combine  the 
individual  scores  of  various  classes.  For  this  purpose  the  tabula- 
tion shown  in  Table  I  (p.  22)  has  been  devised.  In  order  to  save  time, 
teachers  very  frequently  use  this  form  of  tabulation  for  individual 
classes  as  well  and  compute  the  medians  directly  from  these  dis- 
tributions. 


REPRESENTATION  AND  INTERPRETATION  OF  RESULTS  35 

TABLE  VIII 

SUMMARY  OF   RESULTS  IN  EQUATION  AND  FORMULA  TEST,   SERIES  B, 
FORT  SMITH,  ARKANSAS 


Score 

Three-Month 
Group 

Six-Month 
Group 

20 

i 

19 

18 

I 

17 

2 

16 

2 

15 

6 

14 

i 

4 

13 

2 

6 

12 

3 

7 

II 

9 

5 

IO 

10 

9 

9 

26 

7 

8 

22 

ii 

7 

35 

6 

6 

28 

9 

5 

26 

3 

4 

12 

4 

3 

7 

4 

2 

7 

I 

9 

O 

4 

Number  of  Pupils 

2OI 

87 

Median  Score 

7.2 

9-9 

Table  VIII  represents,  in  somewhat  greater  detail,  the  results 
of  one  of  the  tests  in  a  city  high  school  having  several  algebra 
classes.  It  shows  the  distribution  of  the  group  that  has  studied 
algebra  for  three  months,  and  the  distribution  of  the  group  that 
has  studied  algebra  for  six  months.  Such  a  table  indicates  the 
extent  of  variation  within  a  group  and  also  the  excessive  amount 
of  overlapping  that  exists  between  the  two  group  distributions. 

The  facts  in  this  table  may  be  more  strikingly  portrayed  if 
presented  graphically  as  in  Fig.  2.  In  this  graph  the  results  of 
two  groups  are  drawn  upon  the  same  scale,  one  placed  above  the 


FIRST  YEAR  ALGEBRA  SCALES 


other.  In  constructing  these  graphs,  the  -X"-axis  represents  the 
scores  attained  by  the  various  pupils,  and  the  F-axis  the  per- 
centage of  pupils  making  each  score. 

Graphs  of  this  type  furnish  a  most  efficient  means  for  showing : 
(l)  The  wide  range  of  abilities  within  a  group.    (2)  The  scores 


Per  Cent 
of  Pupils 


16 


10 


_r 


Three -Month  Group 
i  (201  Pupils) 


Per  Cent 
of  Pupils 


2    4    6 
Score 


14   16   18 


20 


Six-Month  Group 
(87  Pupils) 


—       _.      _-       18       20 
Score 

FIG.  2.   Distribution  and  Median  Scores,  Equation  and  Formula  Test,  Series  B, 
Fort  Smith  High  School,  Fort  Smith,  Ark. 

most  frequently  made  by  pupils  of  a  group.  (3)  The  extent  to  which 
a  group  falls  short  or  surpasses  the  median  standard  achievement, 
as  indicated  by  the  distance  between  the  median  for  the  group  and 
the  standard  median.  (4)  The  amount  of  progress  made  from  group 
to  group,  as  indicated  by  the  distance  between  the  group  medians. 
(5)  The  amount  of  overlapping  between  the  groups.  In  Fig.  2, 


REPRESENTATION  AND  INTERPRETATION  OF  RESULTS 


37 


this  last  factor,  the  excessive  amount  of  overlapping,  is  undoubt- 
edly the  most  significant.  It  will  be  seen  that  29  per  cent  of  the 
pupils  of  the  three-month  group  do  as  well  as  or  better  than,  the 
median  pupil  in  the  six-month  group,  and  that  about  24  (24.4) 
per  cent  of  the  pupils  of  the  six-month  group  fall  below  the  achieve- 
ment of  the  median  pupil  in  the  three-month  group. 

It  is  sometimes  desirable  to  represent  graphically  the  scores 
attained  by  each  individual  pupil  in  a  given  test.  This  is  done  in 
Fig.  3>  where  the  results  obtained  from  an  entire  class  in  the  Addi- 
tion and  Subtraction  Scale  are  exhibited.  Here  each  pupil's  score 


Score 


Standard 


Class,  lie  di_sn_ 
1 


10 


123 


6      7      6      9     10   11    12  13    14    15    16    17   18   19    20  21    22    23   24 
(Each   line   represents   a  pupil's   ecor«) 


FIG.  3.    Individual  Scores  of  a  Six-Month  Group,  Addition  and  Subtraction 
Test,  Series  A,  Moses  Brown  School,  Providence,  R.  I. 

is  represented  by  a  vertical  line,  the  height  of  the  line  indicating 
the  size  of  the  score.  Thus  the  relative  standing  of  each  and  every 
pupil  is  vividly  portrayed.  It  will  be  seen  that  the  best  student 
in  this  class  does  more  than  five  times  as  well  on  this  test  as  the 
poorest  student. 

When  a  composite  picture  of  the  achievement  of  a  pupil  on  all 
five  tests  is  desired,  a  graph  similar  to  that  in  Fig.  4  is  suggested. 
The  scores  of  five  different  pupils  on  each  of  the  five  different  tests 
are  here  exhibited.  The  short  horizontal  lines  indicate  the  median 
score  of  the  class.  Pupil  A,  it  is  evident,  surpassed  the  median 
scores  of  the  class  in  all  of  the  tests;  pupils  B,  C,  and  D  fell  below 
these  medians  in  one  or  more  tests;  and  pupil  E  did  very  poorly 
in  all  but  the  Multiplication  and  Division  Test,  in  which  he  did 
exceptionally  well.  The  relative  standing  of  the  members  of  a 
class  is  more  readily  determined  from  this  graph  if  pupils  are 
arranged  roughly  in  order  of  excellence  of  achievement. 


3o  FIRST  YEAR  ALGEBRA  SCALES 

In  Fig.  5  the  median  scores  made  in  a  city  high  school  on  all 
of  the  tests  in  1920  are  compared  with  the  scores  made  in  the  same 
school  in  1919  and  with  the  standard  scores.  It  is  at  once  evident 
that  the  median  scores  made  in  1920  are  all  well  above  the  stand- 
ard but  not  quite  as  high  as  the  1919  median  scores. 

A  most  interesting  graph  of  results  from  the  same  school  is  re- 
produced in  Fig.  6.  These  data  were  obtained  during  the  school 
year  of  1920-21,  and  clearly  illustrate  the  advantage  of  homo- 
geneous grouping  of  pupils.  At  the  beginning  of  the  year  all  first 
year  algebra  pupils  were  divided  into  three  sections  on  the  basis 

5srr 


;  -• 

=•-. 

r 

-j 

.s 

r. 

" 

JS 

:r_ 

- 

-- 

JS 

-_ 

- 

- 

| 

I     Z 

', 

4 

i 

] 

1 

2 

4 

E 

! 

J 

7 

i 

I 

] 

1 

3 

< 

5 

1    2 

345 

Letters'represent  different  pupils;  perpendicular  lines,  pupil's  individual  scores  in  (i)  Addition 
and  Subtraction,  (2)  Multiplication  and  Division,  (3)  Equation  and  Formula,  (4)  Graph,  (s) 
Problem,  tests  respectively;  horizontal  cross  bars:  —  class  median;  -  -  -  standard  median. 

FIG.  4.     Individual  Scores  Attained  by  Pupils  from  a  Six-Month  Group  on  All 
Tests,  Series  A,  Moses  Brown  School,  Providence,  R.  I. 

of  results  obtained  with  the  Otis  Intelligence  Tests.  The  pupils 
making  the  highest  grades  in  the  intelligence  test,  29  in  all,  were 
placed  in  one  group  and  covered  a  year  and  a  half  of  work  in 
algebra  in  nine  months.  Twenty-two  pupils  were  placed  in  the  slow 
section  and  about  70  in  the  normal  group.  The  bright  group  was 
tested  at  the  end  of  six  months  and  the  other  two  groups  at  the 
end  of  nine  months.  The  graph  indicates  that  the  normal  group, 
covering  the  usual  amount  of  ground,  made  a  satisfactory  showing 
on  all  the  tests;  and,  moreover,  that  the  bright  group,  at  the  end 
of  six  months,  as  shown  by  the  results,  possessed  the  ability  to 
solve  the  algebra  exercises  that  was  far  superior  even  to  that 
represented  by  the  nine-month  standard. 


REPRESENTATION  AND  INTERPRETATION  OF  RESULTS      39 
20. 

16. 


10 


°A  d.    and     Mult,    and       Equat 
Sub.  Div.  and 

Form. 


lij 


Problem    Graph 


Scores.  1919^3  Scores,  19£0 

Standard  Scores  •• 

FIG.  5.    Comparison  of  Median  Scores  Attained  in  1919  and  1920, 
Series  B,  Wellington  High  School,  Wellington,  Kans. 


20- 

• 

„ 

16- 

| 

-    1 

• 

^ 

| 

J5  

\ 

ml 

In 

Add.   and 

Mult,   and       Equat.    and           Graph             Problem 

Sub.                      Div.                  Form. 

Standard  Scores  •••       Scores  of  Bright  Group  at  £nd  of  Six 
MonthSBza      Scores  of  Normal  Groups      Scores  of  Slow  Grouper^ 

FIG.  6.    Comparison  of  Median  Scores  Attained  by  Bright  Group,  Median  Group, 

and  Slow  Group  in  All  of  the  Tests,  Series  B,  Wellington  High  School, 

Wellington,  Kans. 


40  FIRST  YEAR  ALGEBRA  SCALES 

9.    ANALYSIS  OF  ERRORS 

There  is  a  tendency  on  the  part  of  teachers  and  administrators 
to  use  standard  tests  merely  for  the  purpose  of  determining  de- 
grees of  attainment.  Large  numbers,  the  majority  of  the  teachers 
perhaps,  consider  the  net  results,  as  shown  by  individual  and  class 
scores,  as  the  all-important  objective  in  the  use  of  standardized 
tests.  Such  knowledge  of  total  scores  achieved  is  valuable,  but  it 
is  merely  the  first  step  in  the  process  of  securing  greater  efficiency 
in  instruction  through  the  use  of  standard  testing  devices.  Tests 
have  other  values  which  are  much  more  significant  to  the  class- 
room teacher.  They  should  be  used  much  more  extensively  to 
reveal  weaknesses  in  teaching  and  to  aid  in  the  diagnosis  of  dif- 
ficulties encountered  by  pupils. 

In  the  analysis  of  the  results  of  a  test,  the  types  of  problems 
causing  special  difficulty  are  usually  quite  readily  disclosed,  but 
it  is  a  much  more  intricate  matter  to  determine  with  any  degree 
of  precision  the  mental  processes  that  may  be  responsible  for  the 
various  errors. 

Three  studies  on  the  analysis  of  errors  most  frequently  made  in 
algebra  are  here  reported  in  Tables  IX,  X,  and  XL 


TABLE  IX 

CLASSIFICATION  OF  443  ERRORSS  MOST  FREQUENTLY  MADE  BY  PUPILS  OF  THE 
FORT  SMITH  (ARK.)  HIGH  SCHOOL  IN  THE  EQUATION  AND  FORMULA  TEST, 
KjL*Ju%  ^^  *4  L       SERIES  B 

1.  Performing  wrong  operation  in  solving  for  unknown: 

Ex.  7.  —  z  =    6  or         —  2  =  6 

3  3 

12  2 

2    = 2=6 

3  3 

2.  Error  in  sign  in  transposition: 

Ex.  4.    5^  +  5     =  61  —  30 
5a  -  30  =  61  +  5- 

3.  Simple  arithmetical  errors: 

Ex.  3.    3*  =  9  -  3 
3*  =  9- 

s  Fort  Smith  Survey:   Classification  of  Errors  in  Algebra,  made  under  the  direction 
of  A.  M.  Jordan,  of  the  University  of  Arkansas. 


ANALYSIS  OF  ERRORS  41 

4.  Error  in  using  the  four  fundamental  operations  of  algebra  : 

Ex.  5.    7»  —    3«  =  12  —  4 

low  =    8. 

Ex.  8.    c  —  2  (3  —  4c)  =  12 
c  —  2  —  6  -  8c  =  12. 

5.  Adding  denominators  in  addition  of  fractions: 


3       4       2 

JL-  JL 

7        2* 

6.  Incomplete  solution: 

Ex.  12.    4;y  +  3?  =  30 

7y   =  30. 

Ex.  18.    —  4*  =  2 

-  x  =  \. 

7.  Error  in  sign  in  division: 

Ex.  6.     -  32  =  -  6 

2    =    —  2. 

8.  Error  in  copying: 

Ex.  14.   7m  —  $n  =  12 
-jn    —  3«  =  12. 


9.   Using  exponent  for  coefficient: 

Ex.  19.   p2  —  5p  -  50 
-  3P  =  50. 

10.  Error  in  substituting  the  value  of  the  unknown  in  a  formula 

Ex.  1 6.    Area  of  a  triangle  =  \  bh. 
Find  area  when  b  =  10  ft. 
and  h  =    8  ft. 
Area  of  triangle  =  5  X  4  =  20. 

11.  Solving  for  wrong  unknown  in  a  formula: 

Ex.  17.  RM  =  EL,  solve  for  M. 
EL 


FIRST  YEAR  ALGEBRA  SCALES 


»• 

4. 


Performing  the  wrong  operation  in 
solving  for  unknown 

Error  in  sign  in  transposition 

Simple  arithmetical  errora 

Error  in  using  the  four  fundamental 
operations  in  algebra 

Adding  denominators  in  addition  of 
fractions 


6.  Incomplete  solution 

7.  Error  in   sign   in  division 

8.  Error  in  copying 

9.  Using  exponent  for* coefficient 

10.  Error  in  substituting  the  value  of 

the  unknown   in  a  formula 

11.  Solving  for  the  wrong  unknown  in  a 

formula 

12.  Unclassified 


443      100* 


FIG.  7. 

Distribution  of  443  Errors  made  by  Three  and  Six-Month  Groups  on 
Equation  and  Formula  Test,  Series  B 


ANALYSIS  OF  ERRORS  43 

TABLE  X 

SUMMARY  OF  ERRORS  MADE  BY  PUPILS  IN  36  WISCONSIN  HIGH  SCHOOLS  •  ON 
FOUR  TESTS,  SERIES  B,  MARCH  AND  APRIL,  1921 

Per  Cent 
of  Total 
No.  Errors 
ADDITION  AND  SUBTRACTION  TEST 

Failure  to  deal  with  parentheses  correctly      38 

Failure  to  write  the  denominator  of  a  fractional  answer      24 

Wrpng  process  (adding  instead  of  subtracting,  for  example) 18 

Writing  the  sum  of  the  numerators  for  a  new  numerator  and  the  sum  of 

the  denominators  for  a  new  denominator 4 

Errors  whose  cause  could  not  be  discovered 

Miscellaneous  errors    .  8 


Total ioo 

MULTIPLICATION  AND  DIVISION  TEST 

Mistakes  in  dealing  with  exponents 37  <~ 

Using  the  wrong  process 14  <- 

Mistakes  in  the  use  of  signs 13  ^ 

Mistakes  in  factoring 11^ 

Failure  to  deal  with  parentheses  correctly      6  ** 

Errors  whose  cause  could  not  be  discovered 1 1 

Miscellaneous  errors 8 

Total ioo 

EQUATION  AND  FORMULA  TEST 

Failure  to  change  signs  when  transposing 28  >- 

Multiplying  by  the  coefficient  of  the  unknown  in  order  to  solve    ....  18  i~ 

Use  of  wrong  process      10  *" 

Mistakes  in  solving  literal  formulae 

Errors  in  dealing  with  parentheses 7  u 

Mistakes  in  substitution  in  formulae 7*" 

Finding  one  root  only  in  solving  a  quadratic 7  ** 

Dividing  by  the  numerator  of  the  coefficient  of  the  unknown  in  order  to 

solve 6  ^ 

Errors  which  could  not  be  explained 5 

Miscellaneous  errors .  4 

Total ioo 

j 
PROBLEM  TEST 

Errors  due  to  ignorance  of  fundamental  relationships  (Those  in  length, 

breadth,  thickness,  and  volume  for  example) 52  ** 

Errors  due  to  misreading  the  problem 41  u* 

Errors  which  could  not  be  explained 7 

Total ioo 

•Osburn,  W.  J.:  Survey  of  Algebra  Instruction  in  Wisconsin  High  Schools.  These 
pupils  had  studied  algebra  for  six  months.    The  total  number  of  pupils  tested  varied 

from  1055  to  1635.     Some  of  the  schools  did  not  find  it  possible  to  give  all  four  tests. 


44  FIRST  YEAR  ALGEBRA  SCALES 

TABLE  XI 

CLASSIFICATION  OF  ERRORS1"  MOST  FREQUENTLY  MADE  BY  PUPILS  OF  THE  FORT 
SMITH  (ARK.)  HIGH  SCHOOL  IN  THE^PROBLEM  TEST,   SERIES  B 

(w*^Lt*JA 

1.  Incorrect  operation  indicated,  usually  due  to  failure  to  comprehend  the 
problem.   Caused  55  per  cent  of  the  errors. 

Prob.  I :  If  a  coat  costs  x  dollars,  how  much  will  3  coats  cost? 
Answer:  x  —  3. 

2.  Conditions  of  the   problem   apparently   understood,   but  the   work  left 
in  incomplete  form.   Caused  15  per  cent  of  the  errors. 

Prob.  7.  The  total  number  of  circus  tickets  sold  was  836.  The  number 
of  tickets  sold  to  adults  was  136  less  than  twice  the  number  sold  to 
children.  How  many  were  sold  of  each? 

Answer:  x  =  No.  of  children's  tickets  sold 

2x  —  136  =  No.  of  adults'  tickets  sold. 

L-,     3.   Failure  to  comprehend  the  problem,  perhaps  due  to  confusion  and  to  use 
of  technical  terms.  Caused  10  per  cent  of  the  errors. 

Prob.  6.  The  width  of  a  basket  ball  court  is  20  feet  less  than  its  length. 
What  is  the  length  and  width  of  the  court  if  the  perimeter  (distance 
around)  is  240  feet? 

Equation:  x  —  x  —  20  =  240. 

L,   4.   Inverting  the  order  of  terms  in  subtraction  and  in  division.   Caused  5  per 
cent  of  the  errors. 

Prob.  2:  A  man  is  m  years  old:  how  old  was  he  r  years  ago? 
Answer:  r  —  m. 

Prob.  5.  The  distance  from  Chicago  to  New  York  is  980  miles.  If  a 
train  runs  v  miles  an  hour,  what  is  the  time  required  for  the  run? 

Answer:      -^- 
980 

<--     5.   Simple  arithmetical  errors.    Caused  5  per  cent  of  the  errors. 

6.   Attempt   to   solve   problems   containing   two  unknowns   with   only  one 
equation  containing  the  two  unknowns.    Caused  2  per  cent  of  the  errors. 


10.    VALUE  OF  FIRST  YEAR  ALGEBRA  SCALES 

These  scales  may  be  used  by  teachers  for  three  distinct  and  very 
useful  purposes.  They  may  be  used  (a)  to  indicate  attainment,  (b) 
to  measure  progress,  and  (c)  to  diagnose  difficulties. 

Scales  which  increase  in  difficulty  by  approximately  equal  steps 
furnish  a  most  reliable  objective  means  for  determining  the  actual 

10 Fort  Smith  Survey:  Classification  of  Errors  in  Algebra,  made  under  the  direction 
of  A.  M.  Jordan,  of  the  University  of  Arkansas. 


VALUE  OF  FIRST  YEAR  ALGEBRA  SCALES  45 

achievement  of  a  student  or  a  group  of  students.  Any  one  of  the 
scales  may  be  used  for  this  purpose,  though  the  Equation  and 
Formula  Scale  is  perhaps  to  be  preferred,  since,  as  previously  stated, 
it  is  a  more  comprehensive  test.  It  is  well  to  keep  in  mind  also,  in 
this  connection,  that  a  low  median  class  score  is  not  always,  nor 
even  quite  generally,  due  to  poor  instruction.  Any  one  or  a  com- 
bination of  several  causes  may  be  operating  to  keep  a  class  score 
down.  It  is  the  duty  of  the  teacher,  however,  to  study  these  causes 
and  to  learn  which  ones  are  affecting  the  efficiency  of  the  instruc- 
tion, in  order  that  proper  remedial  measures  may  be  applied.  This 
it  is  possible  to  accomplish  with  a  much  greater  degree  of  certainty 
when  the  teacher  knows  the  actual  standard  of  achievement  a  class 
has  attained.  Such  knowledge  furnishes  the  teacher  with  a  fact 
basis  upon  which  to  proceed  and  a  motive  with  which  to  operate. 

The  extent  of  progress  made  by  a  class  can  be  quite  scientifically 
measured  by  submitting  the  same  scale  to  a  class  at  intervals  of 
about  three  months.  Teachers  should  be  cautioned  very  specifi- 
cally, however,  not  to  do  any  drill  work  upon  the  exercises  or 
problems  appearing  in  the  scales.  If  it  is  feared  that  some  of  the 
practice  effect  may  survive,  it  is  suggested  that  another  scale  in 
the  same  series,  or  the  same  scale  in  a  different  series,  be  used  for 
the  second  test.11  The  most  desirable  method  of  measuring  progress, 
very  naturally,  would  be  to  have  another  parallel  series  of  scales 
similar  and  equal  in  difficulty  to  those  of  Series  A,  and  it  is  to  be 
hoped  that  such  a  series  will  soon  be  constructed. 

For  diagnostic  purposes  the  scales  of  Series  B  have  been  found 
to  be  more  serviceable.  They  offer  a  richer  variety  of  exercises  and, 
therefore,  a  greater  number  of  type  processes.  Hence,  a  more  com- 
plete analysis  of  the  mistakes  made  by  students,  and  the  difficulties 
they  encounter,  is  made  possible. 

Finally,  it  must  be  stated  emphatically  that  these  are  primarily 
power  tests  and  as  such  should  never  be  used  for  purposes  of  drill. 
Furthermore,  with  the  time  limits  as  now  fixed,  they  are  speed  tests 
to  a  limited  extent  only.  If  a  pure  speed  test  is  desired,  the  Stand- 
ard Tests12  devised  by  Dr.  H.  O.  Rugg  could  be  used  to  advantage. 
These  would  be  particularly  useful  in  determining  whether  a  class 
has  had  sufficient  drill  upon  the  fundamentals. 

11  See  suggestions  for  using  the  scales  in  rotation,  p.  9. 

12  See  School  Review,  October,  1917,  25:546. 


46  FIRST  YEAR  ALGEBRA  SCALES 

II.    BIBLIOGRAPHY 

I.  Material  in  Periodicals  and  Books: 

Cawl,  Franklin  R.:  Practical  Uses  of  an  Algebra  (Hotz)  Standard  Scale.  School 
and  Society,  July  19,  1919. 

Harris,  Eleanora:  Study  of  the  Hotz  First  Year  Algebra  Scales  and  the  Rugg- 
Clark  Standard  Algebra  Tests.  Master's  Dissertation,  University  of  Chicago, 
1919. 

Hobbs,  James  B.:  Results  from  Giving  the  Hotz  First  Year  Algebra  Scale  Tests  to  a 
Six-Eight  Month  Group.  School  and  Society,  October  16,  1920. 

Hotz,  Henry  G.:  First  Year  Algebra  Scales.  Contributions  to  Education,  No.  90, 
Teachers  College,  Columbia  University,  New  York  City,  1918.  (This  is  a  tech- 
nical monograph  giving  a  detailed  account  of  the  statistical  methods  employed 
in  the  derivation  of  the  scales.) 

Ramsey,  J.  W. :  A  Study  of  the  Intelligence  of  Paragould  (A  rk.}  High  School  Students. 
Master's  Dissertation,  Peabody  College  for  Teachers,  1921. 

II.  Material  in  School  Surveys  and  Educational  Reports: 

Survey  of  the  Virginia  Public  Schools.    Part  II.    Educational  Tests. 

Survey  of  the  Reading  (Pa.)  High  School. 

Survey  of  the  School  System  of  Mount  Holly,  N.  J. 

Survey  of  the  Public  Schools  of  Hackensack,  N.  J. 

Survey  of  the  Philadelphia  Schools. 

Survey  of  Public  Education  in  North  Carolina. 

Biennial  Report,  1916-1918,  State  Department  of  Public  Instruction,  Madison, 
Wisconsin. 

Report  of  Division  of  Educational  Tests,  1919-20,  Bureau  of  Educational  Re- 
search, University  of  Illinois,  Urbana,  111. 

The  scales  may  be  secured  from  the  publishers:  Bureau  of 
Publications,  Teachers  College,  Columbia  University,  New  York 
City,  or  from  the  following  distributing  centers:  Public  School 
Publishing  Co.,  Bloomington,  Illinois;  Bureau  of  Educational 
Measurements  and  Standards,  State  Normal  School,  Emporia, 
Kansas. 


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